You might consider some combination of lens + converter on a crop body camera rather than your very nice 5DIII, or just crop the images your 5DIII takes in post which is probably even better than using a crop body. Probably not the advice you want, but it might be worth jotting down some notes to see what combinations that would give you and in the end could likely cost you less than a 70-200 IS which was your suggested price range.
Something like getting a 2x converter plus a crop body, with your existing 70-200 brings you to a crop body equivalent of a 224-640 @5.6 for example. Again, using the same combination on your 5DIII and cropping the image to the same composition will probably produce comparable if not better results? Again probably not what you want, just a thought.